The most boring model (part 1)

Consider the most boring model for a quantity \(y(x)\) \[ \frac{dy}{dx} = \beta \] with initial condition \(y(0)=\alpha\). The rate of change of \(y\) as a function of \(x\) is constant.

What function of \(t\) has slope \(\beta\) for every \(x\)? The linear function \[ y(x) = \alpha + \beta x . \] Have you seen this before?

The most boring model (part 2)

Recall the linear “regression” model \[ y = \alpha + \beta x + \epsilon \] where \(\epsilon\) is a mean-zero error term.

If \(x\) is a treatment, and we regard \(y(x)\) as the potential outcome when treatment is set to \(x\), then the “treatment effect” is the change in \(y\) induced by a unit change in \(x\), \[ \frac{dy}{dx} = \beta \] This is one reason that some people (economists) think of causal effects as derivatives of outcome functionals.

Birth process

Let \(y(t)\) be the number of bacteria in a dish at time \(t\). Suppose at time \(t=0\), there are \(y(0) = y_0\) bacterium. Each bacterium is immortal, and divides into two with rate \(\lambda>0\). The number of bacteria at time \(t\) is governed by the differential equation \[ \frac{dy}{dt} = \lambda y(t) \] with initial condition \(y(0)=y_0\). The solution is \[ y(t) = y_0 e^{\lambda t} \] You can verify this by computing \(dy/dt\). The dynamics of this system obey “exponential growth”. The speed of this growth is governed by \(\lambda\).

Birth process

Death process

The “death” process is the same, but we let \(\lambda=-\mu<0\). Then \[ \frac{dy}{dt} = -\mu y(t) \] with initial condition \(y(0)=y_0\), which has solution \[ y(t) = y_0 e^{-\mu t} \] and \(y(t)\) decreases/decays “exponentially” toward zero.

Logistic growth I

Recall the exponential growth model \[ \frac{dy}{dt} = \lambda y(t) \] with initial condition \(y(0)=y_0\). Its solution is \(y(t) = y_0 e^{\lambda t}\). As \(t\to\infty\), \(y(t)\to\infty\). This unbounded growth may be unrealistic. For example, resource constraints may prevent cells from dividing indefinitely.

Suppose \(K\) is the “carrying capacity” of the medium, and \(\lambda\) is the intrinsic gowth rate as above. Consider the model \[ \frac{dy}{dt} = \lambda y(t)\left( 1 - \frac{y(t)}{K}\right) \] with initial condition \(y(0) = y_0\). This is just like the pure birth model, except there is a negative quadratic term.

Logistic growth II

\[ \frac{dy}{dt} = \lambda y(t)\left( 1 - \frac{y(t)}{K}\right) \]

\(y(t)\) is constant when \(dy/dt=0\), which occurs whe either \(y(t)=0\) or \(y(t)=K\). Therefore

• If \(y(t) < K\), then \(dy/dt>0\) and the population size increases toward \(K\).
  
• If \(y(t) > K\) then \(dy/dt<0\) and the population size decreases toward \(K\).

In fact, the solution can be written down precisely: \[ y(t) = \frac{K y_0}{y_0 + (K-y_0)e^{-\lambda t}} \]

Logistic growth

Logistic growth: SI model

Consider a population of individuals where every individual is susceptible or infected with an infectious disease. Let \(S(t)\) be the proportion of susceptibles, and let \(I(t)\) be the proportion of infectives. Then the system is characterized by \[ \frac{dI}{dt} = \beta S(t) I(t) \] where \(S(t) + I(t) = 1\). We can rewrite this model as a logistic growth model with \(K=1\) and \(\lambda=\beta\) and \(y_0=I(0)\) above.

SIR model

Consider a population of \(N\) individuals where every individual is susceptible or infected or recovered. Let \(S(t)\) be the number of susceptibles, \(I(t)\) be the number of infectives, \(R(t)\) the number of recovered. Then the system is characterized by

\[ \begin{aligned} \frac{dS}{dt} &= -\beta S(t) I(t) \\ \frac{dI}{dt} &= \beta S(t) I(t) - \gamma I(t) \\ \frac{dR}{dt} &= \gamma I(t) \end{aligned} \]

with initial condition \(S_0\), \(I(0)\), \(R(0)\) and conservation equation \(S(t)+I(t)+R(t)=N\).

SEIR model

Now suppose there is a latent period following infection that delays onset of infectiousness. Individuals in this latent state are exposed, but not yet infectious.

\[ \begin{aligned} \frac{dS}{dt} &= -\beta S(t) I(t) \\ \frac{dE}{dt} &= \beta S(t) I(t) - \delta E(t) \\ \frac{dI}{dt} &= \delta E(t) - \gamma I(t) \\ \frac{dR}{dt} &= \gamma I(t) \end{aligned} \]

SEIR model with vital dynamics

Let’s incorporate births and deaths and model the absolute number of individuals in each compartment instead of the proportion. Let \(N(t) = S(t) + E(t) + I(t) + R(t)\). Let \(\mu\) be the per-person birth rate, and let \(\nu\) be the per-person death rate.

\[ \begin{aligned} \frac{dS}{dt} &= \mu N(t) - \nu S(t) -\beta S(t) I(t) \\ \frac{dE}{dt} &= \beta S(t) I(t) - (\nu + \delta) E(t) \\ \frac{dI}{dt} &= \delta E(t) - (\nu + \gamma) I(t) \\ \frac{dR}{dt} &= \gamma I(t) - \nu R(t) \end{aligned} \]

SEIRS (loss of immunity)

\[ \begin{aligned} \frac{dS}{dt} &= -\beta S(t) I(t) + \sigma R(t) \\ \frac{dE}{dt} &= \beta S(t) I(t) - \delta E(t) \\ \frac{dI}{dt} &= \delta E(t) - \gamma I(t) \\ \frac{dR}{dt} &= \gamma I(t) - \sigma R(t) \end{aligned} \]

Example: SIR models for policy

Krebs et al. (2019)

Example: HIV care pipeline

\[ \begin{aligned} \frac{dD}{dt} &= \lambda_D I(t) - (\lambda_L + \mu) D(t) \\ \frac{dL}{dt} &= \lambda_L D(t) - (\lambda_A + \mu) L(t) \\ \frac{dA}{dt} &= \lambda_A L(t) - (\lambda_S + \mu) A(t) \\ \frac{dS}{dt} &= \lambda_S A(t) - \mu S(t) \\ \end{aligned} \]

In one-way queueing models you can think of the time spent in each compartment as the inverse of the rate of exit.

Adapted from Gonsalves et al. (2017).

Example: HIV care pipeline

Gonsalves et al. (2017)

Pharmacokinetic models

Let \(G(t)\) be the amount of drug in the GI tract, \(B(t)\) in the bloodstream, and \(O(t)\) in the organ of interest. Let \(I(t)\) be the known input dynamics of drug administration (oral, IV, etc).

\[ \begin{aligned} \frac{dG}{dt} &= I(t) - \lambda_G G(t) \\ \frac{dB}{dt} &= \lambda_G G(t) - \mu_B(t) - \lambda_T B(t) \\ \frac{dO}{dt} &= \lambda_T B(t) - \mu_O O(t) \end{aligned} \]

The \(\lambda\) parameters represent rates of transition to the next compartment, and \(\mu\) parameters represent rates of clearance.

Incorporating heterogeneity: SIR with two susceptible types

You can often incorporate heterogeneity by splitting compartments.

\[ \begin{aligned} \frac{dS_1}{dt} &= -\beta_1 S_1(t) I(t) \\ \frac{dS_2}{dt} &= -\beta_2 S_1(t) I(t) \\ \frac{dI}{dt} &= (\beta_1 S_1(t) + \beta_2 S_2(t)) I(t) - \gamma I(t) \\ \frac{dR}{dt} &= \gamma I(t) \end{aligned} \]

General-purpose ODE solver: lsoda

Lots of ODEs can’t be solved analytically. For (well-behaved) systems that have a unique solution, it is often possible to solve for \(y(t)\) numerically, without having any calculus-style analytic insight into the structure of the solution. There are very fast computer algorithms for doing this.

We will focus on methods implemented in standard software packages.

library(deSolve)

you will need to install this.

For help, look at ?deSolve

Example: SIR solution

library(deSolve)
sir.model <- function(t, x, params) {
    S <- x[1]
    I <- x[2]
    R <- x[3]
    with(as.list(params), {
        dS <- -beta * S * I
        dI <- beta * S * I - gamma * I
        dR <- gamma * I
        dx <- c(dS, dI, dR)
        list(dx)
    })
}

Example: SIR solution

times <- seq(0, 120, by = 1)
params <- c(beta = 0.4, gamma = 1/7)
xstart <- c(S = 9999/10000, I = 1/10000, 
    R = 0)
out <- as.data.frame(lsoda(xstart, times, 
    sir.model, params))

Example: SIR solution

plot(out$time, out$I, ylab = "Population proportion", 
    xlab = "Time", type = "l", bty = "n", 
    ylim = c(0, 1), col = "red", lwd = 2)
lines(out$time, out$S, col = "green", lwd = 2)
lines(out$time, out$R, col = "blue", lwd = 2)
legend(100, 0.5, c("S(t)", "I(t)", "R(t)"), 
    lty = 1, col = c("green", "red", "blue"))

Step 1: specify the model

sir.model <- function(t, x, params) {
    S <- x[1]
    I <- x[2]
    R <- x[3]
    with(as.list(params), {
        dS <- -beta * S * I
        dI <- beta * S * I - gamma * I
        dR <- gamma * I
        dx <- c(dS, dI, dR)
        list(dx)
    })
}

First, we see that we have created a function sir.model, which is a funciton of time t, a model state x, and parameters params. The state x is a three-component vector corresponding to the S, I, and R states of the model.

Specifying the model

sir.model <- function(t, x, params) {
    S <- x[1]
    I <- x[2]
    R <- x[3]
    with(as.list(params), {
        dS <- -beta * S * I
        dI <- beta * S * I - gamma * I
        dR <- gamma * I
        dx <- c(dS, dI, dR)
        list(dx)
    })
}

The with function takes a list of parameters and does an action (contained in the curly brace block) using those parameters in memory. The action here is definition of the derivatives dS, dI, and dR, and putting these together in a list.

Set up the times, parameters, and starting values

times <- seq(0, 120, by = 1)
params <- c(beta = 0.4, gamma = 1/7)
xstart <- c(S = 9999/10000, I = 1/10000, 
    R = 0)

This part just sets up some times where we will evaluate the model, some parameters, and starting conditions.

Running lsoda

Here is the most important part

out <- as.data.frame(lsoda(xstart, times, 
    sir.model, params))

The lsoda function takes these arguments and solves the system for every time in times. We store the result as a data frame instead of as a list. Here is what out looks like

head(out)

##   time         S            I            R
## 1    0 0.9999000 0.0001000000 0.000000e+00
## 2    1 0.9998541 0.0001294936 1.638852e-05
## 3    2 0.9997947 0.0001676659 3.760126e-05
## 4    3 0.9997179 0.0002170761 6.506202e-05
## 5    4 0.9996191 0.0002805687 1.003539e-04
## 6    5 0.9994911 0.0003628204 1.460811e-04

ODEs made easy

1. Write the components (“compartments” or “types” or “states”) in the time-dependent system you want to model. Examples: \(S(t)\), \(I(t)\), \(R(t)\)
   
2. Write the time-derivatives of the components as a function of time, and other compartments. Example: \(dS/dt\).
3. Write the initial conditions for all the components. Example: \(S(0)=99\), \(I(0)=1\).
4. Simulate the dynamics of the system

ODEs are most interesting when there are interactions between compartments.

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